General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Aug 22, 2018

Limit analysis of solid reinforced concrete structures

Larsen, Kasper Paaske

Published in:Computational Technologies in Concrete Structures

Publication date:2009

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Larsen, K. P. (2009). Limit analysis of solid reinforced concrete structures. In Computational Technologies inConcrete Structures (pp. 154)

Limit Analysis of Solid Reinforced Concrete Structures

Kasper Paaske Larsen*1,2) Peter Noe Poulsen2) Leif Otto Nielsen2)

1) Ramboll Denmark, Teknikerbyen 31 DK-2830 Virum, Denmark2) Department of Structural Engineering, DTU, Kgs. Lyngby 2800, Denmark1) kas@byg.dtu.dk

AbstractRecent studies have shown that Semidefinite Programming (SDP) can be used effectively for limitanalysis of isotropic cohesive-frictional continuums using the classical Mohr-Coulomb yield cri-terion. In this paper we expand on this previous research by adding reinforcement to the modeland a solid element for lower bound analysis of reinforced concrete structures is presented. Themethod defines the stress state at a point within the solid as a combination of concrete- and re-inforcement stresses and yield criterions are applied to the stress components separately. Thismethod allows for orthotropic reinforcement and it is therefore possible to analyze structures withcomplex reinforcement layouts. Tests are conducted to validate the method against well-knownanalytical solutions.

1 IntroductionThe first reinforced concrete structures came about in the last of part of the 18th century andconcrete is today the most widespread construction material. In the last part of the 1970ties thefirst numerical solutions for the determination of the load bearing capacity of reinforced con-crete structures was presented by (Fredsgaard & Kirk, 1979; Damkilde & Kirk, 1981). Sincethen, formulations for the load bearing capacity has been presented for a wide range of structuralcomponents: beams (Damkilde & Hoyer, 1993a), plates (Damkilde & Hoyer, 1993b), stringers(Damkilde et al., 1994) and disks (Poulsen & Damkilde, 2000). Even though many different el-ement types has been formulated, no reel 3D modeling capabilities existed because the differentelements had been designed separately with little focus on interaction. The disk element (Poulsen& Damkilde, 2000) was combined with bars and beams but still only in 2D. In 2007 (Nieblinget al., 2007) formulated a 3D beam element for reinforced concrete based on a 3x3 zone model.The zone model utilized the previously established yield surfaces for disks in the zones and therebycircumvented the formulation of the total yield surface for the beam with all of its special cases.Similarly a shell element was established on the basis of zone models, once again utilizing thedisk yield surfaces. The combination of the beam element and the shell element made it possibleto analyze 3D structures. For evaluation of the assumptions made in the zone models of the 3Dbeam element and the shell element a demand for a solid 3D modeling tool has evolved. In thearea of geotechnics, solid 3D elements have been established e.g. for Mohr-Coulomb materials(Krabbenhøft et al., 2008). The 3D formulation of Mohr-Coulomb materials leads to SemidefiniteProgramming (SDP) problems, see e.g. (Krabbenhøft et al., 2008). Today only a few optimization

1

programs has the capability to solve SDP problems and the present work uses sedumi (Sturm &Sturm, 1999; Pólik, 2005) for this task and Yalmip (Löfberg, 2004) is used as interface to thesolver.For reinforced concrete the solid modeling in 3D of the reinforcement has to be addressed. Thiscan be handled either by modeling of each reinforcement bar with realistic dimensions or by sim-ply adding a tensile capacity in the direction of the reinforcement corresponding to the amountof reinforcement (Nielsen, 2008). The second choice is the basis for the solid 3D modeling ofreinforced concrete in the present article. This makes geometric modeling of the structure muchsimpler but it also introduces some assumptions concerning the transfer of stresses between theconcrete and the reinforcement. The example verifies that the material model works as intendedand the results are compared with some analytical solutions.

2 Finite Element Formulation of the Lower Bound TheoremThe lower bound theorem of limit analysis states that a safe and statically admissible stress dis-tribution will not be able to cause collapse in the structure. If we let all external load componentsbe proportional to the load factor λ > 0 then will all solutions in which λ ≤ λp, where λp is thecollapse load, be a lower bound solution. In practice we are not just interested in any lower boundsolution i.e. 0 ≤ λ ≤ λp but rather the solution closest to the collapse load i.e. min(λp −λ ). Thisleads to an optimization problem which is often difficult, if not impossible, to solve manually.In this paper we present a finite element framework for limit analysis of solid reinforced con-crete structures. The three dimensional stress field is approximated using four node tetrahedralelements with linear stress distribution. Because of the convex properties of the yield criterion, astrict lower bound solution can be obtained by enforcing the yield criterion at the nodes.

2.1 Statically admissible stress distributionAs prescribed by the lower bound theorem, the chosen stress distribution must be statically admis-sible. For this requirement to be fulfilled when implemented in a finite element system, the internalstress state within each element as well as the inter-element stresses must be in equilibrium andthe static boundary conditions satisfied.

2.1.1 Internal element equilibrium

Let the three dimensional stress state at a point within the volume be defined by the stress tensor

σi j =

σx τxy τxzτyx σy τyzτzx τzy σz

(2.1)

where σi are the normal stresses and τi j for i ̸= j are the shear stresses. The equilibrium equationcan, with application of the symmetric convention for i and j, be written as

λ fi +σ ji, j = 0 (2.2)

2

Figure 1: Inter-element equilibrium is ensured by traction continuity at the interfaces between elements.

where fi are the proportional body forces per unit volume and λ is the load factor. The tensornotation σ ji, j refers to the partial derivative of component ji with respect to direction j.For the four node tetrahedral element with a linear stress field, the partial derivatives becomesconstant and the body force field is then approximated by piecewise constant body forces.

2.1.2 Inter-element equilibrium

Inter-element equilibrium is ensured by requiring traction continuity at the interface between adja-cent elements. Figure 1 shows two tetrahedral elements which shares a common boundary definedby the vertices v2,v3 and v5. If p is a point on the interface and the stress state in each element atthat point is defined by σ p1

i j and σ p2i j respectively, the inter-element equilibrium can be written as

(σ p1i j −σ p2

i j )n j = 0 (2.3)

where n j is the unit normal vector of the plane defined by v2,v3 and v5. As seen here, the stressstate does not have to be continuous from one element to the next as long as the tractions on theshared boundary are equal.

2.1.3 Static boundary conditions

Surface traction equilibrium in global coordinates gives

−λ ti +σi jn j = 0 (2.4)

where t ′i are the proportional surface tractions, n j is the outward unit surface normal and λ isthe load factor. If one or more traction components are supported at a point on the boundary,the associated equations are removed from the system of equations, allowing the stresses to varyfreely within the yield constraints.

3

Figure 2: Solid concrete with orthogonal reinforcement.

2.1.4 Safe stress distribution

A volumetric infinitesimal domain of concrete is reinforced in directions equivalent to the usualCartesian coordinate system as shown on Figure 2. The stresses on the boundary of the infinitesi-mal domain are given by λ (σ0

x ,σ0y ,σ0

z ,τ0xy,τ0

xz,τ0yz) where λ is the load factor. The shear stresses

in the rebars are disregarded and the normal stresses are smoothed over the cross section areaperpendicular to each of the three directions. These are denoted σsi where i ∈ {x,y,z}. Stressequilibrium on the boundary gives

λσ 0x = σsxAsx +σcx λσ0

y = σsyAsy +σcy λσ0z = σszAsz +σcz

λτ0xy = τcxy λτ0

xz = τcxz λτ0yz = τcyz (2.5)

where Asx, Asy and Asz are the reinforcement area per unit area perpendicular to the three axis and(σcx,σcy,σcz,τcxy,τcxz,τcyz) are the concrete stresses. If the reinforcement is not aligned with theCartesian coordinate system, the equivalent orthotropic reinforcement values are used.Since the shear stresses in the reinforcement are disregarded, the yield criterion for the rebar’s canbe formulated as simple upper- and lower bounds on the normal stress components as

fyc,i ≤ σsi ≤ fyt,i i ∈ {x,y,z} (2.6)

where fyc,i and fyt,i are the compression and tensile strengths respectively.

2.2 Modified Coulomb material using Semidefinite ProgrammingThe modified Coulomb criterion is described in (Nielsen, 1999) and is the most commonly usedyield criterion for limit analysis of concrete structures. It consists of the Coulomb sliding failurecriterion combined with a separation criterion. If tensile stresses are positive, the sliding failurecan, in the three dimensional case, be written as

kσc1 −σc3 ≤ fc (2.7)

where σc1 and σc3 are the largest and smallest principal stress respectively. fc is the uniaxial com-pression strength of the concrete and k is a friction parameter. Here is used k = 4 corresponding

4

to a friction angle ϕ = 37◦.The separation criterion can be formulated as

σc1 ≤ fct (2.8)

where fct is the separation strength, which for a typical concrete is equal to the uniaxial tensilestrength.

2.2.1 Semidefinite Programming (SDP)

The application of SDP in limit analysis of Coulomb materials has been described in (Krabbenhøftet al., 2008), (Martin & Makrodimopoulos, 2008) and(Bisbos & Pardalos, 2007).Semidefinite Programming considers the problem of minimizing a linear function of the variablesx ∈ ℜm subjected to a set of matrix inequalities and equality constraints as

minimize cT x (2.9)subject to F(x)≽ 0

Ax = b

where the constraint function

F(x) = F0 +m

∑i=1

xiFi (2.10)

is an affine combination of symmetric matrices. The problem data are the vectors c ∈ ℜm andb∈ℜk, the matrix A∈ℜk×m and the m+1 symmetric matrices F0, . . . ,Fm ∈ℜm×m. The inequalityF(x) ≽ 0 is called a Linear Matrix Inequality or LMI and states that the constraint function ispositive semidefinite, i.e. xtF(x)x≥ 0 for all x∈ℜm. Since positive semidefinite cones are convex,(Vandenberghe & Boyd, 1996), the SDP is a convex optimization problem.Of course if the matrix F ≽ 0 then

F +λF I ≽ 0 (2.11)

for all λF ≥ λmax where λmax is the largest eigenvalue of F .

2.2.2 SDP formulation of the Modified Mohr-Coulomb criterion

As described in section 2.1.4 the stress state at a point consists of a combination of concrete- andreinforcement stresses, Eq.( 2.5). The reinforcement constraints can simply be formulated as linearinequalities based on Eq. (2.6) but the modified Coulomb criterion must be cast as semidefiniteconstraints.The Coulomb criterion in Eq. (2.7) can be written as a combination of two positive semidefinitecones, see (Krabbenhøft et al., 2008; Martin & Makrodimopoulos, 2008; Bisbos & Pardalos,2007). Eq. (2.12) and (2.13) shows this formulation using the concrete material parameters fc andk as posed by (Nielsen, 2008).

σ ci j + kαI ≽ 0 (2.12)

−σ ci j +

(fc

k−α

)I ≽ 0 (2.13)

5

where α is an auxiliary variable, I is the unit matrix and σ ci j is a stress tensor containing the

concrete stresses.The separation criterion given in Eq. (2.8) can be formulated directly as an LMI constraint on theform

−σ ci j + I fct ≽ 0 (2.14)

2.3 Solving lower bound problems using the YALMIP interfaceImplementing the linear constraints posed by the equilibrium constraints and the yield criterionsfor the reinforcement is straight forward since they are part of the general formulation of theSDP problem, Eq. (2.9). Most SDP solvers only allow for a single LMI constraint on a set ofvariables, so the modified Coulomb criterion described in section 2.2.2 proves a bit more difficultto implement. Instead of reformulating the problem to fit the format of a specific solver, it ischosen to use the YALMIP interface, (Löfberg, 2004). YALMIP provides a parametric modelinglanguage that enables developers a high-level model for defining different types of optimizationproblems. One of the main advantages of using YALMIP is that it takes care of all the low-levelmodeling and is designed to obtain as efficient and numerical sound model as possible. Anotheradvantage is that it supports a wide range of solvers, making it possible to switch from one solverto another by simply changing the optimizer settings. YALMIP is implemented as a MatLabTM

toolbox.

3 Numerical ExamplesThis section shows how the numerical lower bound model presented here can be used to solvesome basic examples. There are currently only a few solvers capable of solving SDP problems.Here we use the open source solver SeDuMi, (Sturm & Sturm, 1999; Pólik, 2005) which is im-plemented as a MatLabTM toolbox 1. All default settings are used except for the way SeDuMihandles free variables. By default, SeDuMi places free variables in a quadratic cone which causedstability problems. These problems were resolved by forcing SeDuMi to split the free variablesinstead of placing them in cones.All tests are performed on a Mac Pro Workstation (2 x 2.8 GHz Quad Core Xeon, 6 GB RAM)running Windows XP Pro x64 under Bootcamp.

3.1 Numerical determination of yield condition for a reinforced solidIn the first example, the numerical model will be used to determine the yield conditions for a solidblock of reinforced concrete with various types of reinforcement. For these tests, a cubic blockwith a side length of 5 is modeled using a structured mesh of 3 ·3 ·3 ·24 = 648 elements as shownon Figure 3. The yield surface is a function of the six independent stress parameters in the stresstensor σi j. To simplify visualization, only a section in the σxτxy-plane is determined here. This is

1The 64 bit version of SeDuMi bundled with CVX is used (Grant & Boyd, 2008a; Grant & Boyd, 2008b).

6

Figure 3: Square block of reinforced concrete discretized using a structured mesh.

done by applying cosine- and sine weighted values of σx and τxy to the block

τ0xy = λ sin(v) σ0

x = λ cos(v) (3.1)

were 0 ≤ v ≤ π .The following material parameters are used unless otherwise noted

fc = 1 fct = 0.0 k = 4fyt = 1 fyc = 1 in the x,y,z directions

3.1.1 Isotropic Disc

In (Nielsen, 1999) the yield condition for a reinforced concrete disc is determined based on thelower bound theorem. In this example we will show how the yield surface for a disk can bereplicated using the numerical model. The complete surface is a bit complex, but figure 2.2.11 in(Nielsen, 1999) shows the curve of intersection between the yield surface and the σxτxy-plane, andthe expression for this curve is summarized below

τxy(σx) =

√

Φ fc (Φ fc −σx) if −(1−2Φ) fc ≤ σx ≤ Φ fc√Φ(1−Φ) fc if − fc ≤ σx ≤−(1−2Φ) fc√14 f 2

c −[σx +

(12 +Φ

)fc]2

if −(1+Φ) fc ≤ σx ≤− fc

The isotropic disc is easily modeled by simply removing the reinforcement in the direction per-pendicular to the loaded plane, in this case the z-direction

Φx = Φy = 0.1Φz = 0.0

Figure 4 shows the τxyσx-section of the yield surface for both the analytical and the numericalmodel. As seen from the figure, a very good correlation between the analytical and the numericalmodel is achieved. The difference observed on the figure is caused by the subdivision with whichthe numerical curve is generated.

7

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

σx

τxy

Numerical solutionAnalytical solution

Figure 4: Comparison between the numerical and the analytical yield surface for a reinforced concretedisk. The numerical curve is determined using 100 linearly spaced values of v in the interval 0 toπ .

3.1.2 Isotropic Solid

The previous example showed how the numerical model could be used to recreate a known yieldcondition for an isotropic disc. This example investigates the yield conditions for an isotropicsolid of reinforced concrete, i.e.

Φx = Φy = Φz = 0.1

Figure 5 shows the τxyσx-section of the yield surface as determined by the numerical analysis andcompared with the disc solution, some tri-axial effects are seen. One thing to note is the effectson the uniaxial compression strength, which here is determined to 1.5. This is identical to thetheoretical value which can be found by assuming full utilization of the transverse reinforcement

σsy = σsz = As fy (3.2)

The concrete stress components in the y- and z-direction can be determined from Eq. (2.5) whenσ0

y = σ0z = 0 as

σcy = σcz =−As fy (3.3)

With no shear stresses acting on the cube, the normal stresses are equal to the principal stresses(no off-diagonal elements in the concrete stress tensor). Since the uniaxial compression strengthis greater than the transverse compression generated by the reinforcement, the principal stressesin the concrete material becomes

σc1 = σc2 =−As fy σc3 = σ0x − (− fy)As (3.4)

8

−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

σx

τxy

Solid solutionDisk solution

Figure 5: The yield surface of a reinforced concrete solid determined using the numerical model. Thenumerical curve is determined using 100 linearly spaced values of v in the interval 0 to π .

The uniaxial compression strength can be determined from the sliding criterion given in Eq. (2.7).

σ0x ≤ fc(1+Φ(k+1)) = 1.5 · fc (3.5)

4 ConclusionsWe have shown a method for performing limit analysis of reinforced concrete structures using ageneral finite element framework based on lower bound elements. The method uses SemidefiniteProgramming algorithms for solving the optimization problem posed by the lower bound method.The solver used here (SeDuMi) has proven very efficient and robust for these types of problemsand the use of YALMIP as interface made it very easy to formulate the optimization problems.To simplify modeling of complex structures a material model in which reinforcement and concreteare considered as a homogeneous material. The stress state in this material is defined by a com-bination of concrete- and reinforcement stresses and yield criterions are imposed on these stresscomponents separately. The numerical model has been verified by comparison with well knownanalytical solutions and good correlation has been found between the results. The approximationof considering the reinforcement and concrete as a unified material has also been tested and theresults are very encouraging.

9

ReferencesBisbos, C., & Pardalos, P. 2007. Second-Order Cone and Semidefinite Representations of Material

Failure Criteria. Journal of Optimization Theory and Applications, 134(2), 275–301.

Damkilde, L., & Hoyer, O. 1993a. An efficient implementation of limit state calculations basedon lower-bound solutions. Computers and Structures, 49(6), 953–962.

Damkilde, L., & Hoyer, O. 1993b. An efficient implementation of limit state calculations basedon lower-bound solutions. Computers and Structures, 49(6), 953–962.

Damkilde, L., Olsen, Jørgen F., & Poulsen, Peter N. 1994. A Program for Limit State Analysisof Plane, Reinforced Concrete Plates by the Stringer Method. Bygningsstatiske Meddelelser,65(1), 1–26.

Damkilde, Lars, & Kirk, Jens. 1981. RUPTUS - a program for rupture analysis (in danish). ADB-Udvalget.

Fredsgaard, Søren, & Kirk, Jens. 1979. RUPTUS - et program til brudstadieberegninger. ADB-Udvalget.

Grant, M., & Boyd, S. 2008a (December). CVX: Matlab software for disciplined convex program-ming.

Grant, Michael, & Boyd, Stephen. 2008b. Graph Implementations for Nonsmooth Convex Pro-grams. Recent Advances in Learning and Control, 95–110.

Krabbenhøft, K., Lyamin, A. V., & Sloan, S. W. 2008. Three-dimensional Mohr-Coulomb limitanalysis using semidefinite programming. Communications in Numerical Methods in Engi-neering, 24(11), 1107–1119.

Löfberg, J. 2004. YALMIP : A Toolbox for Modeling and Optimization in MATLAB. In: Pro-ceedings of the CACSD Conference.

Martin, Christopher M., & Makrodimopoulos, Athanasios. 2008. Finite-element limit analysisof Mohr-Coulomb materials in 3D using semidefinite programming. Journal of EngineeringMechanics, 134(4), 339–347.

Niebling, J., Vinther, A., & Larsen, K. P. 2007. Numerisk Modellering af Plastiske Betonkonstruk-tioner. M.Phil. thesis, Department of civil engineering, BYG-DTU.

Nielsen, Leif Otto. 2008 (January). Concrete Plasticity Notes. Tech. rept. Department of CivilEngineering, Byg-DTU.

Nielsen, M. P. 1999. Limit Analysis and Concrete Plasticity. CRC Press.

Pólik, I. 2005. Addendum to the SeDuMi user guide. Version 1.1.

10

Poulsen, Peter Noe, & Damkilde, Lars. 2000. Limit state analysis of reinforced concrete platessubjected to in-plane forces. International Journal of Solids and Structures, 37(42), 6011–6029.

Sturm, Jos F., & Sturm, Jos F. 1999. Using SeDuMi 1.02, a MATLAB toolbox for optimizationover symmetric cones. Optimization Methods and Software, 11(1), 625–653.

Vandenberghe, Lieven, & Boyd, Stephen. 1996. Semidefinite programming. SIAM Review, 38,49–95.

11